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Created 7/1/1996
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Business Administration 130

Brealey and Myers, chapter 8: Introduction to Portfolio Theory

http://www.j-bradford-delong.net/Intro_Finance/BAonethirty8.html

 


Basics:


Not happy with the lecture I gave last time; tried to do to much and gave much-too-quick explanations of important things. So let me back up a bit, and go over the "benefits of diversification" again.


Benefits of Diversification

"The opportunity cost of capital depends on the risk of the project": I've been saying this for three and a half weeks now. But what does it mean? What is the risk of a project? Why should the appropriate cost of capital vary depending on how risky the project is?

Let's start with risk.

 State of the World

 Mega Manufacturing

 Startup Semiconductor

 HH

 +40%

 +10%

 HT

 +10%

 -20%

 TH

 +10%

 +40%

 TT

 -20%

 +10%

 Expected Value...

 EV = +10%

 EV = +10%

 Standard Deviation...

 SD = 21.2%

  SD = 21.2%

What risk-return combination do you get if you put all your money into one stock or the other?

 

But suppose you start mixing one with the other...

 25%M+75%S

 Mega Manufacturing

 Startup Semiconductor

 17.5%

 +40%

 +10%

 -12.5%

 +10%

 -20%

 32.5%

 +10%

 +40%

 2.5%

 -20%

 +10%

 EV = +10%

 EV = +10%

 EV = +10%

 SD= 16.8%

 SD = 21.2%

  SD = 21.2%

 

 0M+1S

.25M+.75S

.5M+.5S

.75M+.25S

1M+0S

 Expected Return

 +10%

 +10%

 +10%

 +10%

 +10%

 Standard Deviation

 21.2%

16.8%

15%

16.8%

21.2%

 


More General Formulas for Variances:

Unpack this formula: what does it mean?


An especially interesting special case:

Suppose we have a number of securities, indexed by i, all of whose returns look something like this:

and suppose that the ui's are uncorrelated with each other and with the market--that their covariance is zero, or is at least not too large.

Now suppose that we take our portfolio, and our portfolio shares xi, and set each xi=1/N: put 1/N of our portfolio's wealth into each of the first N of these securities.

What does our variance look like?

Well, we know that:

And we can substitute in:

Noticing that the first and third terms look awfully familiar:

is the variance of the portfolio:

Now a funny thing happens as N gets large: the second term vanishes: the sigmas stay (roughly) the same size, and you are adding up more (N) of them, but each one is divided by N-squared. So as N approaches infinity, the second term gets small--eventually small enough to ignore...

An even funnier thing happens if we consider the variance of a portfolio made up of a large number of stocks (N large), for which the average beta happens to be one.


Conclusion:

If an investor is doing his or her job--trying to get to a portfolio that has the minimum risk for a given expected return--then "idiosyncratic" risk can be diversified away: by putting all your eggs into many baskets, you essentially eliminate any idiosyncratic risk factors from your portfolio.

Hence a security's "riskiness"--from the point of view of its impact on the overall riskiness of your portfolio as a whole--is summarized by its beta...


Risk and Return

Now let me move on to risk and return proper...

 "State of the World"

Universal
Utility

Mega
Manufacturing

Exciting
Exports

Startup
Semiconductor

 HH

 +20%

 +40%

 +40%

 +20%

 HT

 +0%

 +10%

 +50%

 -20%

 TH

 +10%

 +10%

 -30%

 -20%

 TT

 -10%

 -20%

 -20%

 +20%

Suppose I am choosing portfolios from Mega Manufacturing and Startup Semiconductor:

I get the highest return from Mega Manufacturing, but I can get a lower standard deviation--at not much cost in return--by starting to diversify.

 

Suppose I diversify among the four different stocks:

I can construct a large number of truly, truly putrid portfolios (in fact, by throwing money into the sea I can construct even worse portfolios...)

I can also get outside the frontier of the parallelogram defined by the risk/return characteristics of the individual stocks...

Introducing lending and borrowing:

Choose the portfolio S that just touches the line that goes through the riskfree-rate point and lies entirely to one side of the feasible portfolio set.

Then borrow (and lend) until you get to the risk-return characteristics you want...

Capital asset pricing model...